Paso 8: Determinación de los espesores de las sobrecapas
5. Recapeo o refuerzo asfaltico
2.1.5 COSTOS EN PROYECTOS DE CARRETERAS
The properties of the basic unobservable single server queue were dis- covered by Edelson and Hildebrand [47]. They adopted the first eight assumptions of Naor’s observable model as listed in§2.1, and added the following modifications.
Assumption 10 changes as follows:
At the time a customer’s need for service arises, he irrevocably either joins the queue or balks. It is not possible for him to observe the queue length before making this decision.
Assumption 8 is relaxed as follows:
The service discipline is strong and work-conserving.1
As in the observable model, a customer who joins the queue imposes negative externalities on others and therefore individual optimization leads to excessive congestion unless the queue is regulated.2 This prop- erty will be formally proved below.
1For example, the service discipline may be FCFS, LCFS (with or without preemption), EPS,
or random order. Under all these disciplines, the expected waiting time is given in (1.4).
2In a more general context, Hardin [64] showed that if each individual uses a common resource
to maximize his own utility, then the equilibrium has excessive use of the resource.
46 TO QUEUE OR NOT TO QUEUE
1.1.
Equilibrium
We start by evaluating the customers’ behavior in equilibrium when an admission fee of size p is imposed and the potential arrival rate is Λ. There are two pure strategies available for a customer: to join the queue or not to join. A pure or mixed strategy can be described by a fraction q, 0 ≤ q ≤ 1, which is the probability of joining. Given that the admission fee is p, we denote the equilibrium probability of joining by qe(p), and the corresponding equilibrium (effective) arrival rate by
λe(p). Of course, λe(p) =qe(p)Λ< µ. We denote the expected waiting
time when the (effective) arrival rate is λ < µ by W(λ) = µ−1λ . (For
λ≥ µ, define W(λ) = ∞.) This function is continuous and monotone increasing. The net benefit for a customer who joins the queue is the value of service, R, minus the full price, p+CW(λ). We distinguish three cases:
p+CW(0) ≥ R. In this case, even if no other customer joins, the net benefit of a customer who joins is non-positive. Therefore, the strategy of joining with probabilityqe(p) = 0 is an equilibrium strat-
egy and no other equilibrium is possible. Moreover, in this case, not joining is a dominant strategy.
p+CW(Λ)≤R. In this case, even if all potential customers join, they all enjoy a non-negative benefit. Therefore, the strategy of joining with probability qe(p) = 1 is an equilibrium strategy and no other
equilibrium is possible. Moreover, in this case, joining is a dominant strategy.
p+CW(0) < R < p+CW(Λ). In this case, if qe(p) = 1 then a
customer who joins suffers a negative benefit. Hence, this cannot be an equilibrium strategy. Likewise, if qe(p) = 0, a customer who joins
gets a positive benefit, more than by balking. Hence, this too can- not be an equilibrium. Therefore, there exists a unique equilibrium strategy whereqe= λe(Λp) and whereλe(p) solvesCW(λe(p)) =R−p.
SubstitutingW(λ) = µ−1λ, we obtain the expressions given in Table 3.1.
Unobservable queues 47 1 1 Best response qe(p) q 45o
Figure 3.1. The best response vs. the joining probability
Case λe(p) qe(p) W(λe(p)) Λ≤µ− C R−p Λ 1 1 µ−Λ 0≤µ− C R−p ≤Λ µ− C R−p µ−RC −p Λ R−p C µ− C R−p <0 0 0 1 µ
Table 3.1. The equilibrium strategy
Remark 3.1 Suppose that the probability of joining is q: if q < qe(p) then the unique best response is 1, ifq =qe(p) then any strategy between
0 and 1 is a best response, and ifq > qe(p) then the unique best response
is 0. Since the best response is a monotone non-increasing function of the strategy, the model is of the ATC type. Figure 3.1 depicts the best response function and the equilibrium point.
1.2.
Social optimization
We now turn our attention to social optimization. Let the socially optimal joining probability be q∗, and let the socially optimal joining
48 TO QUEUE OR NOT TO QUEUE rate beλ∗ whereλ∗ =q∗Λ. Then,
λ∗ = arg max
0≤λ≤Λ{λ[R−CW(λ)]}.
Since W(λ) is strictly convex, the function to be maximized is strictly concave and has a unique maximum. SubstitutingW(λ) = µ−1λ we get that the solution
µ− s Cµ R = arg max0≤λ<µ λR−Cλ 1 µ−λ
is optimal as long as it is in [0,Λ]. The fact that the solution is nonneg- ative follows from the assumption thatRµ≥C. Thus, if Λ≥µ−qCµR
thenλ∗ =µ−qCµR. Otherwise, λ∗= Λ. Let SU 3 be the social welfare
under the optimal arrival rate λ∗. Table 3.2 summarizes the optimal joining strategy. Case λ∗ q∗ W(λ∗) S U Λ≥µ− q Cµ R µ− q Cµ R µ−pCµR Λ q R Cµ √ Rµ−√C2 Λ≤µ− q Cµ R Λ 1 1 µ−Λ Λ R− C µ−Λ
Table 3.2.The socially optimal strategy
It follows from the assumptionRµ ≥C thatλe(0)≥λ∗. Thus, as in
the case of observable queues, individual optimization leads to queues that are longer than are socially desired. This gap can be corrected by imposing an appropriate admission fee, as discussed in the next section.
Balachandran and Srinidhi [23] observed that ifλe(0)<Λ then
1−λ∗ µ 2 = 1−λe(0) µ .
Unobservable queues 49 Remark 3.2 Assume thatλ∗<Λ. Consider a tagged customer who is given the lowest possible priority so that he is served only when there are no other customers in the system. In particular, his service may be preempted (and resumed later from the point of interruption). The change in the tagged customer’s priority has no effect on social welfare. However, with this change, the tagged customer imposes no externalities. By (1.6) his expected waiting cost when the arrival rate equals λ∗ is
C µ
1−λµ∗−2, which by Table 3.2 equalsR. Hence, ifλ=λ∗, the tagged customer is indifferent between joining and not; if λ < λ∗, he prefers joining; and ifλ > λ∗, his choice is not to join. Thus, as expected, when
λ = λ∗ the tagged customer who imposes no externalities behaves in the socially desired way. This principle will be used in§4.5 to present a decentralized way for optimally controlling an unobservable queue. Remark 3.3 We assumed that the service duration follows an exponen- tial distribution. However, unlike in the observable case, the same qual- itative results are obtained for any service distribution. Balachandran and Srinidhi [23] examined the sensitivity of the solution to uncertainty, as reflected by the second moment x2 of the service time distribution.
They concluded that both λ∗ and λe(0), as well as the ratio λ
∗
λe(0), are
monotone decreasing inx2. The latter property means that:
The need to control the queue becomes more critical as the uncer- tainty measured by the variance of the service requirement, increases.
1.3.
Profit maximization
We consider now a monopolistic server that sets a profit-maximizing admission feepm. A monopoly does not leave a positive customer sur-
plus, since in such a case the admission fee can be increased without re- ducing the arrival rate. Therefore,pm+CW(λ) = R.The monopoly’s
problem is to maximizeλpm subject to 0≤λ≤Λ and
pm =R−CW(λ). (3.1)
Recall that the social objective is to maximize the total welfare of the server and the customers, which is λp+λ[R−CW(λ)−p]. The termλpcancels, reflecting the assumption that social utility is additive so that from social point of view the admission fees are merely transfer payments that have no effect on social welfare. Hence, the social problem is to maximizeλ[R−CW(λ)] subject to 0≤λ≤Λ. By (3.1):
50 TO QUEUE OR NOT TO QUEUE λ∗ pm R−C µ R−pRC µ Λ
Figure 3.2. Monopoly prices vs. rate of arrival
The socially optimal arrival rateλ∗ can be induced by an appropriate admission fee, which also maximizes total profit. Whenλ∗ <Λ this fee equals
pm =p∗ =R−CW(λ∗) =R−
s
CR µ .
When λ∗ = Λ, the profit maximizer chooses the maximum fee which induces this rate, that is,pm=R−µ−CΛ. A social planner would choose
this fee, or any smaller fee, since any such choice induces the same optimal arrival rate, Λ.
Chen and Frank [33] observed that:
pm ismonotone non-increasing in Λ (see Figure 3.2).
Thus, an increase in demand may lead to a reduction in price! This may seem counterintuitive, but can be explained as follows: when the arrival rate increases the expected waiting time increases, and customers are inclined to pay less for service. In other words:
Since the quality of the goods depends inversely on the demand, the price needs to be reduced when demand increases.
Edelson and Hildebrand also discussed an extension of their model in which the server imposes a two-part tariff. For a given inspection feea customer may choose to inspect the queue length and then choose
Unobservable queues 51 between balking and paying anadmission feeto join the queue. For any given admission fee, the profit-maximizing inspection fee coincides with the customer’s expected gain from inspecting the queue. Thus again, all of the customer’s surplus goes to the server and the server chooses socially optimal fees.
Remark 3.4 The model does not allow reneging. However, this as- sumption is redundant if we assume that the queue remains unobservable also after joining. This is because the expected residual waiting time is non-increasing with the time already spent in the queue (see §5.2). In anM/M/1 queue, due to the memoryless property of the waiting time, this value is constant, whereas when reneging is exercised by a positive fraction of the customers, it is strictly decreasing with the time in the queue.
Remark 3.5 Joining with probability qe(p) is an equilibrium strategy also in a LCFS-PRobservablequeue without reneging. The reason is that the length of the queue is unimportant to the new arrival. However, as shown by Tilt and Balachandran [168] and Hassin and Haviv [73], this is not the unique SPE.